Combining predictions with errors

Question

If i were to try to combine two predictions, such as:

Test 1: $70\% \pm 10\%$ sure that $x$ is between $1.5$ and $2.5$
Test 2: $65\% \pm 10\%$ sure that $x$ is between $1.5$ and $2.5$

combination of test $1$ and two find $XX\% \pm YY\%$ sure that $x$ is between $1.5$ and $2.5$.

Is it a coincidence that 1.5 and 2.5 appear in both cases, or is that indeed the general intention. — Ami Tavory, Aug 02 '17 at 14:53
Possible duplicate of [Combining probabilities/information from different sources](https://stats.stackexchange.com/questions/155817/combining-probabilities-information-from-different-sources) — Tim, Aug 02 '17 at 14:53
the range is intentionally consistent, I am comfortable with how I will combine the ranges if they vary, it's how to accurately combine the percentages with uncertainties I can't do. running two tests and getting the same result should, in theory, make you more confident and reduce the error, I think? — G.Bruce, Aug 02 '17 at 14:55
Hi @Tim, I have read the question you have referenced there and think this is a different situation. This has an associated error, which I cannot figure out how to carry through. I also don't think the confidence would be 67.5%, because running the second test and getting the same result should increase how confidently I can state that x lies between 1.5 and 2.5. Thanks. — G.Bruce, Aug 02 '17 at 14:57
Have you considered this as a meta-analysis problem? I may have misunderstood your intentions though. — mdewey, Aug 02 '17 at 16:05
It isn't clear to me that this is a duplicate. But I think it would help if you explained what "70% plus or minus 10% sure that $x$ is between 1.5 and 2.5" means - this isn't the usual way that confidence intervals are written for example. What does the "plus or minus 10%" refer to? — Silverfish, Aug 02 '17 at 17:22
Thanks Silverfish, I understand this is an odd format. If you had someone very confident telling you they were 70% sure someone was perfectly healthy, but knew they had an unreliability due to a lack of information of approximately 10 percent. This Is a simplified version of events of course, but I have been told a system is approximately 70 percent accurate, with an uncertainty in the accuracy of 10% — G.Bruce, Aug 02 '17 at 17:31
Forgive me if that's still unclear, it's hard to explain properly without getting stuck into the details — G.Bruce, Aug 02 '17 at 17:35
You can if it makes the problem easier, swap the range of 1.5 to 2.5 to an integer value, as I can get a confidence In an integer too — G.Bruce, Aug 02 '17 at 17:41

Combining predictions with errors

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